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Calculation of the connective constant for self-avoiding walks (Vol. 44 No. 5)

A typical self-avoiding walk of 100 million steps on the square lattice, generated using the pivot algorithm.

Self-avoiding walks are walks on a lattice, which are not allowed to self-intersect. Despite the apparent simplicity of the self-avoiding walk model, it is an important model of polymers, and over the past 60 years it has resisted all attempts to find an exact solution.

One of the basic features of self-avoiding walks is the number of walks for a given number of steps. The number of walks grows exponentially with length, and the rate of exponential growth, called the connective constant, is a quantity of fundamental interest.

Using a novel divide and conquer Monte Carlo algorithm, the number of self-avoiding walks on the simple cubic lattice for selected lengths of up to 38 million steps were estimated to high precision. For instance, the number of walks with 606 207 steps is 7.7 × 10406 535! Using these estimates the connective constant was found to be 4.684 039 931 ± 0.000 000 027, which is significantly more accurate than estimates obtained via alternative methods.

A key open question is whether similarly powerful enumeration methods can be found for other models in statistical mechanics.

Nathan Clisby, ‘Calculation of the connective constant for self-avoiding walks via the pivot algorithm’, J. Phys. A: Math. Theor. 46, 245001 (2013)
[Abstract]